Partial quotients and representation of rational numbers

It is shown that there is an absolute constant $C$ such that any rational $\frac bq\in]0, 1[, (b, q)=1$, admits a representation as a finite sum $\frac bq=\sum_α\frac {b_α}{q_α}$ where $\sum_α\sum_ia_i(\frac {b_α}{q_α})<C\log q$ and $\{a_i(x)\}$ denotes the sequence of partial quotients of $x$.

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